capacity commitment versus flexibility

CAPACITY COMMITMENT VERSUS FLEXIBILITY

MARCEL BOYER

CIRANOE´cole Polytechnique

andUniversite de Montreal

Montreal, Canada H3A 2A5

MICHEL MOREAUX

ERNA-INRAGREMAQ

andIDEI, Universite de Toulouse I

31000 Toulouse, France

We show how technological flexibility choices and equilibrium configurations(both simultaneous and sequential duopoly) depend on six industry character-istics. Low market volatility combined with intermediate market size favorsinflexible technologies; large values of either volatility or size favor flexibletechnologies; low or intermediate values of both favor the coexistence of flexibleand inflexible technologies. The possibility of a flexibility trap exists in indus-tries of low volatility and intermediate size. Entry prevention can sometimesbe achieved by inflexible technologies or flexible technologies, depending onthe industry characteristics.

1. Introduction

We are concerned in this paper with the emergence of new technologiescommonly referred to as flexible manufacturing systems (FMSs)1 and inparticular with the ‘‘stylized fact’’ that Japan seems to be investing in

We would like to thank a referee and an associate editor for their comments; JacquesCremer, Drew Fudenberg, Bentley Macleod, Jean-Jacques Laffont, Jean Tirole, MihkelTombak, and Lars-Hendrik Roller for helpful comments and discussions; and FCAR(Quebec), SSHRCC (Canada), CNRS (France), and INRA (France) for financial support.We remain, of course, solely responsible for the content of this paper.

1. They are also referred to as computer-controlled (or -integrated) manufacturing(CAM/CIM).

q 1997 Massachusetts Institute of Technology.Journal of Economics & Management Strategy, Volume 6, Number 2, Summer 1997, 347–376

348 Journal of Economics & Management Strategy

such technologies almost four times more than the United States perdollar of GNP. According to Tchijov (1992), Japan has about 24% ofFMSs in operation around the world, and the United States 16%; theratio of the Japanese GNP to the American GNP is about 40%. The FMSproduction systems being adopted at an increasing rate throughout theworld have the following main economic characteristics: they requirea higher investment cost than conventional technologies, but they allowfor significant reductions in setup costs, allowing more product flexibil-ity and more volume flexibility (reduced minimum production runs)in lead time, in unit variable cost, and in average batch size.2 These newtechnologies are likely to lead not only to very different competitiveenvironments but also to significant modifications of growth patterns.3

FMSs are indeed changing dramatically the production process andthe internal organization of firms as well as their market environmentand their relations with suppliers and customers.

Given the major changes that a FMS represents, the evaluation ofsuch investments has been less than well understood by the engineering,

2. On the basis of the relatively restrictive criteria of the International Institute forApplied Systems Analysis in Austria (see Tchijov, 1992), there were in 1989 about 800FMSs in operation in 26 countries: among others, Japan has about 24% of those, theUnited States 16%, the UK 11%, France 9.7%, and Canada 0.5%; in terms of number ofproducts, 30% of those FMSs are used to produce no more than 10 products, 48% between11 and 100, 19% between 100 and 1000, and 3% over 1000; in terms of batch size, 32%are used to produce batches that are on average smaller than 10 units, 34% between 11and 50, 29% between 50 and 1000, and 5% over 1000. Prudent estimates indicate thatthe introduction of a FMS has allowed an average reduction in lead time (lag betweenorder and delivery) by a factor of 2 to 3, an average reduction in setup time (the timespent to reset the equipment for a product change) by a factor of 10, an average reductionin personnel by a factor of 2 to 3, and a reduction in unit cost by a factor of 1.25 to 1.5.More recent developments in FMSs increase those factors. Milgrom and Roberts (1990)provide other examples of the impact of the adoption of these modern technologies: asurvey of the aerospace and other high-precision industries revealed that 8.2% of allbatches were of size 1 and 38% of size 16 and less; an Allen-Bradley plant producingelectric controls can now switch production among its 725 products with an averagesetup time of 6 seconds, and it usually fills orders the day after they are received andships products that same day by air express; General Motors engineers could in 1988set plant equipment to produce pilots of the 1989 models during the weekend and resetthe equipment in time for the Monday morning production of the 1988 models, whereasthe same operation used to take weeks; General Electric has reduced the lead time forcircuit-breaker boxes from three weeks to three days, which allowed a reduction in backorders from 60 days to 2 days; Caterpillar’s modernization program has been accompa-nied by a doubling of its product line.

3. See Boyer (1991): Growth patterns in the future are likely to be based on a bettermatching between products and preferences rather than on more units per capita, becauseof the shift from economies of scale towards economies of scope allowed by FMSs. Eatonand Schmitt (1994) have shown that this switch from scale to scope has also importantimplications for market structure and competition policy.

Capacity Commitment versus Flexibility 349

finance, and accounting personnel of the typical firm.4 The main difficul-ties in evaluating FMS projects revolve around the proper identificationof the determinants of the value of more flexibility, of better quality con-trol and better product reliability, and of reduced lead time in produc-tion, incorporating in particular the value of organizational (incentives)restructuring possibilities, and the value of the strategic advantages (anddisadvantages) of an investment in FMSs. A leitmotif of the engineeringliterature on the subject is that this state of affairs generates a bias to-wards rejecting such investments because major potentially favorableelements are either misunderstood or simply left out in the profitabilityevaluation, in particular the strategic impact of more flexibility and thechange in the firm’s cost of capital and capital structure.5

Different theories or models could explain, more or less convinc-ingly and more or less directly, the above stylized fact regarding the rela-tive FMS investment levels in Japan and the United States.6 In one suchmodel, the working of financial markets differs in such a way that thelevel of monitoring of entrepreneurs by financiers is higher in Japan;given the presence of adverse-selection and moral hazard in financialmarkets, this allows for a lower cost of capital in Japan; since technologi-cal investments in FMSs are typically long-term investments, the modelpredicts that the level of such investments will be larger in Japan. A re-lated model makes the case that for cultural reasons Japanese managersand stockholders take a longer-run approach (longer payback periods)in evaluating technological investments; since many of the benefits ofFMSs are long-term benefits, the prediction follows. A third model restson the belief or fact that successful investments in FMSs require not onlya concerted organizational transformation of the firm itself but also coor-dinated technological investments and organizational changes by sup-pliers and by clients; since this coordination is more easily reached in thewell-integrated Japanese industrial groups (the keiretsus using a kanbansystem7) than in the more loosely integrated American industrial clus-ters, the model predicts larger FMS investments in Japan.

Our results suggest a different explanation based on the strategicinteractions between the firms combined with the specific characteris-tics of an industry. Given that technology decisions are typically longer-

4. For convincing overviews of the problems, see Gerwin (1982); Lederer and Singhal(1988), who provide an extensive list of references to the engineering literature; andMensah and Miranti (1989).

5. Although this paper is a paper on technological flexibility, much of its content canbe applied mutatis mutandis to organizational flexibility.

6. Some of those were not developed with reference to the above ‘‘stylized fact,’’ butthey may be reinterpreted in this context.

7. See Aoki (1988, pp. 208–223).


term decisions than production ones, we propose a two-stage duopolymodel in which firms choose in stage 1 their respective levels of techno-logical flexibility and choose in stage 2 their production levels. Weshow that there exist very reasonable cases, which we characterize, inwhich the Nash equilibrium in technology is asymmetric, one firmbeing flexible and the other inflexible, even if the two firms are com-pletely symmetric: they have the same information and the same finan-cial, cultural, and coordination possibilities. We also show that thereexist very reasonable cases, which we characterize, in which the firmssuffer in equilibrium from too much flexibility: they experience a flexi-bility trap, a form of prisoner’s dilemma. We then introduce a variantof the above model by assuming that the long-term technology choicesare made sequentially with the second-mover (follower) firm observingthe choice of the first mover (leader) before deciding on its own technol-ogy. We characterize the conditions or industry characteristics underwhich different asymmetric equilibrium configurations will emerge: an(ƒ, i ) configuration with the first mover flexible (ƒ) and the secondmover inflexible (i ), or an (i, ƒ ) configuration. We show that being thefirst mover is at least as profitable as being the second mover for allequilibrium configurations. The stylized observation that differenttechnologies, in particular different levels of flexibility, may coexist inan industry equilibrium has no pathological aspects: such asymmetricconfigurations in technological choices are a reasonable outcome oftechnological competition in many industries. Our results suggest that,under similar levels of managers’ rationality or competence, similarcapital markets, and similar information structures, an important factorin explaining why the Japanese have been investing more in FMS thanthe Americans could be the relative positioning of the Japanese andAmerican firms in different industries.

Although the literature on technological flexibility is rather largeand covers many fields from engineering, operations research, optimalcontrol, human-resources management, and production managementto economics and game theory, most contributions have consideredbasically a decision-theoretic perspective8; few authors have explicitly

8. The best example may be Milgrom and Roberts (1990), who aimed at developingan optimizing model of the firm capable of generating the following characteristics ofthe development of modern manufacturing methods: a replacement of mass productiontechnologies by flexible multiproduct ones making use of advanced computer-controlledequipment and new organizational designs and stressing quality and quick responsesto market conditions. Their context is a decision-theoretic context, rather than a strategicone, involving a significant drop in some key input costs (such as the costs of collecting,analyzing, and disseminating data, the cost of product design and development, andthe cost of flexible manufacturing equipment), significant complementarities within the


considered and studied the strategic aspects of technological flexibilitychoices, and nobody, to our knowledge, has directly tried to explainin an explicit model of strategic behavior the stylized fact at the rootof the present paper. The closest in spirit is Vives (1989, 1993), whoconsidered two types of uncertainty: prior uncertainty and posterioruncertainty measured by the variability in posterior beliefs updatedfrom an imperfect signal. We will consider in this paper a concept ofuncertainty similar to the first one in conjunction with a perfect signalof demand level, and therefore a case of maximal variability in posteriorbeliefs.9 Vives showed that more prior uncertainty raises the value offlexibility by raising both the flexibility value and the commitmentvalue of the technological parameter value. Vives then showed thatalthough an increase in posterior-belief variability always increases thevalue of commitment, it need not increase the value of flexibility, andhence may lead to reduced technological flexibility. An improvementin the precision of the signal (inducing more variable posterior beliefs)may reduce expected profits when the cost of flexibility is low andtherefore may induce firms to invest less in flexibility. This apparentparadox can be explained as follows. When the cost of flexibility is low,firms tend to be very flexible; in that case, an increase in the variabilityof posterior beliefs may reduce expected profits because the high flexi-bility in the industry implies a reduced variance in prices, and hence inexpected profits. In Vives’s analysis, the distinction between flexibility,chosen for efficiency reasons, and size (average-cost-minimizing pro-duction level), chosen for commitment or strategic purposes, is blurredby the fact that he works with one-parameter (the technological param-eter referred to above) cost functions. We will show that increases in

group of activities that form the elements of a firm’s strategy (multiple positive interac-tions between production factors and between those factors and marketing and organiza-tional variables), and finally, important nonconvexities in the adoption of modern manu-facturing and organization, a typically nonmarginal decision. Milgrom and Roberts showthat the clustering of technological and organizational advances in some firms is noaccident. It is rather the result of profit-maximizing strategies based on the exploitationof the extensive complementarities that exist in marketing, manufacturing, engineering,design, and organization.

9. More precisely, Vives compared two equilibria: the open-loop equilibrium, inwhich firms choose simultaneously both their respective technological parameters andtheir production decision functions, and the subgame perfect equilibrium, in which thefirms compete in a two-stage framework where at the beginning of the second stage ofthe game, the flexibility positions previously chosen by the firms are unveiled and ob-served. Vives considered that the choice of the ‘‘technological parameter’’ in the open-loop equilibrium is a pure flexibility choice, while its choice in the subgame perfectequilibrium is made for both flexibility and strategic reasons. The difference betweenthe technological parameter levels in the two equilibria is defined as the strategic orcommitment value of the technological parameter.


prior uncertainty need not lead to more flexible technologies when weconsider technologies with an explicit parameter for flexibility and anexplicit parameter for size. Moreover, we will show that flexibility perse has important strategic implications and may indeed be used in anentry-preventing strategy.

The paper is organized as follows. In Section 2, we introduce thebasic model and we discuss the related literature on the subject. Section3 is devoted to the characterization of the best reply functions for flexi-bility and inflexibility. We characterize in Section 4 both the Nash andthe Stackelberg equilibria and perform some comparative statics exer-cises on the impact of changes in the industry parameters on the natureof the equilibrium. In Section 5, we look at technological flexibility andinflexibility as means to deter entry when such deterrence is possible.In the conclusion, we come back to the stylized fact mentioned at thebeginning and discuss the empirical implications of our results, boththe lessons learned and the predictions made.

2. A Model of Technological Flexibility

George Stigler (1939) pioneered the analysis of technological or costflexibility. He stated that firms in general have to make a choice amongdifferent equipment, giving rise to different cost configurations—forexample, a cost function that has a relatively wide flat bottom, and acost function that can attain a lower minimum average cost at the ex-pense of steeply rising average cost as production moves away fromthe most efficient scale of production.10 In that spirit, let us characterizea technology by the two parameters (g , x ), where g represents thedegree of inflexibility and x is the level of capacity (average-cost-mini-mizing production level). The strategic value of technological flexibility

10. See Boyer and Moreaux (1989) for a review of the general definitions of flexibilityproposed by Stigler (1939), Marshak and Nelson (1962), and Jones and Ostroy (1984).Technological flexibility is quite different from another concept of flexibility, which wemay call flexibility in timing. In the latter context, an agent remains flexible by postponinga decision in order to benefit from improved information over time. The strategic aspectof such flexibility in timing refers to two different phenomena. First, to the irreversibilityoccurring when a decision, taken with less than full information, nevertheless commitsan agent or a society to an irreversible future set of actions; see Henry (1974), Freixas andLaffont (1984), and Dixit and Pyndick (1993). Second, to the relative value of commitmentversus flexibility, the former corresponding to a decision taken before the uncertainty islevied and the latter to a decision taken after the resolution of uncertainty; see Spencerand Brander (1992) and Sadanand and Sadanand (1995). The present paper is somewhatrelated to the latter strand of the literature on flexibility in timing insofar as we considerat least implicitly the value of the commitment associated with inflexibility versus thevalue of easier adaptation to changing markets associated with flexibility.


comes from the possibility of influencing the behavior and choices ofcompetitors, including eventually the decision to enter or not. We con-sider a duopoly (possibly with a competitive fringe), which is not onlythe simplest possible framework in which such strategic considerationscan be analyzed, but also the industry setting most conducive to stra-tegic interactions. To concentrate on these strategic aspects of flexibility,we will assume that the level of capacity is exogenous. To make thosestrategic aspects as explicit as possible, we will consider two differentmarket structures: a Nash setting in which the long-run technologicalchoices are made simultaneously, and a Stackelberg setting in whichthey are made sequentially with the second-mover firm choosing itstechnology after observing the choice of the first-mover firm. In bothcases, the short-run decisions on production will be assumed to bemade simultaneously once technological choices are determined andobserved by both firms. Hence we have in mind a two-stage model: along-run stage 1 in which firms choose their technologies simultane-ously or sequentially, and a short-run stage 2 in which firms choosetheir marketed quantities simultaneously.

This two-stage formulation will in general give rise to multipleequilibria in stage 2 and discontinuities in profit functions; this reflectsimportant underlying phenomena (shutdown of production, bank-ruptcy, etc.), but makes the analysis more intricate and the results,obtainable by numerical simulations, less intuitive. As a step towardsavoiding those difficulties without missing the basic intuitions and re-sults on the flexibility-versus-inflexibility technological choice problem,we will assume in this paper that the firms can only choose betweentwo levels of flexibility, either perfect flexibility (g 4 ƒ) or perfectinflexibility (g 4 i ). Let us explain what we mean by these two extremepossibilities.

The choice between a flexible technology and an inflexible tech-nology rests in part on efficiency or cost considerations and in part onstrategic considerations.11 We can identify the fundamental cost differ-

11. Kulatilaka and Marks (1988) consider a context where firms with high-flexibilitytechnologies are more capable of switching between factors of production in order toadapt more efficiently to input price changes; we directly use the cost function thatimplicitly incorporates those considerations. Fine and Pappu (1990) consider retaliatorypunishment strategies that firms can deploy when a competitor enters their home market.These strategies are made credible by investing in product-flexible technologies. Clearly,the existence of such technologies may increase competition and reduce the firms’ profitsunless punitive strategies can be enforced (in a repeated-game fashion): the firms maybe caught in a prisoner’s dilemma and therefore could be better off without those flexibletechnologies. Our model includes two-stage nonrepeated competition and volume-flexi-ble technologies; we show the existence of a flexibility trap, and we characterize thewhole map of technological flexibility equilibria as a function of industry characteristics.


ences between flexibility and inflexibility as follows: flexible technolo-gies require a larger investment cost; inflexible technologies are moreefficient (lower average variable cost) for production levels close tocapacity; flexible technologies have a lower setup cost incurred whena new production run starts. In order to fully incorporate the importantdifferences in terms of production costs and to stress the commitmentadvantage an inflexible technology may confer, we will model an in-flexible production process as one that can be operated at a given fixedlevel (at capacity) to produce a given fixed quantity q 4 x, which isthen sold by the firm.12 The cost trade-off between choosing a maximalflexibility level g 4 ƒ and choosing a maximal inflexibility level g 4i is captured by the following three characteristics that we will assumefor the cost function. First, the investment cost H of a flexible technologyis larger than that of an inflexible technology, the latter being set at 0.Second, a flexible technology has no setup cost and a constant marginalcost of production c, while an inflexible technology has a setup costequal to sx and an average direct cost of production equal to 0 for q [$0, x} but infinite otherwise. We will assume for simplicity that bothtechnologies attain the same minimum average variable cost, that is, c4 s. Hence

H ( i ) 4 0 and H (ƒ) 4 H . 0,

C(q ) 4 hcq if g 4 ƒ0 if g 4 i and q4 0,sx if g 4 i and q4 x,` if g 4 i and q[6 $0, x},

c 4 s

We will assume that the firms produce a homogeneous productand that the demand is linear:

p 4 max $0, a 1 b (q L ` q F )}.

Our model is therefore a model of volume flexibility rather than prod-uct flexibility,13 although the two types are intimately related through

12. Admittedly, this is an extreme assumption. It can be relaxed in different ways atthe cost of blurring the argumentation, but the main thrust of the results remains valid.See Lecostey (1994) and Boyer and Moreaux (1996).

13. Roller and Tombak (1990, 1993) consider product flexibility in a differentiatedproduct model in which firms move simultaneously in choosing under perfect informa-tion (no uncertainty) their respective technologies and outputs. Two types of technologiesare feasible: a dedicated technology, which can be used to produce a single product, anda flexible technology, which can be used to produce multiple products. The authorscharacterize the subgame perfect equilibria and show that flexible technologies are morevaluable the larger the size of the market is, the smaller the differential investment cost


the reduced fixed cost that a FMS implies.14 We will suppose that thereis uncertainty in demand in the following sense15: although b is as-sumed to be known with certainty, the parameter a (a measure of thesize of the market) is assumed to be a random variable with probabilitydistribution function F (a ) known by both firms:

a [ [a , a ], with variance V and mean m .

To stress even more the strategic character of technological flexi-bility choices, we will assume that the value of a becomes known atthe beginning of every short-run production period; in other words,the uncertainty of demand is levied after the long-run decisions butbefore the short-run decisions. The two firms, assumed to be risk-neu-tral, will choose their flexibility levels to maximize their expectedprofits.

The advantages and disadvantages of flexibility and inflexibilitystem here from three elements: first, a flexible technology requires alarger investment outlay; second, flexibility allows for an easier adapta-tion to changing levels of demand; third, inflexibility has a commitmentvalue insofar as it means a smaller interval (a singleton here) in whichthe firm will select its quantity produced in stage 2, if it produces atall. In order that an inflexible firm be always better off producing and

is between a flexible and an inflexible technology, the more concentrated the industryis, and the more differentiated the products are. The last effect may be interpreted asfollows. Consider a firm with a dedicated technology in market A. If products A and Bare sufficiently bad substitutes, the firm may switch to a flexible technology allowing itto enter into market B, and therefore increasing competition in B, without affecting toomuch the market conditions in A. If the products are sufficiently good substitutes, theincreased competition in market B will reduce the possible profits in market A. This islikely to develop into a prisoner’s dilemma for the firms in the industry and thereforelead to the emergence of inefficient equilibria with too much investment in flexible tech-nologies. In that context, flexible technologies are never adopted if the products areperfect substitutes. But casual observation of markets or industries in which FMS invest-ments have been undertaken indicates that many of those markets and industries arequite competitive with similar product lines (for instance electronics, machine tools,heavy machinery). By introducing uncertainty in demand, we will characterize the sub-game perfect equilibria with homogeneous products in which flexible technologies arepresent and play a major strategic role.

14. See Gerwin (1993) for a more elaborate classification of flexibility into differentconcepts of mix, changeover, modification, volume, rerouting, material, and responsive-ness flexibilities.

15. If x were chosen endogenously, flexibility would be useless in the absence ofsome variability in demand (in the form either of seasonal patterns in demand or ofgenuine uncertainty), because firms would choose a capacity giving the minimum aver-age cost of production in stage 2. With x exogenous, flexibility may be profitable evenwithout variability in demand, because the flexible technology allows a lower productioncost ex post for production levels away from x.


selling q 4 x than shutting down, we will make the following assump-tion16 on a :

a $ 2bx ` s (which implies that m $ 2bx ` s ).

Hence, an inflexible firm will produce and sell the quantity q 4 x (itsreaction function is constant at x ).

This model representation will allow us to characterize explicitlythe strategic value of technological flexibility choices. The structure ofthe model is such that the solution can be obtained by solving first forthe production choices of the firms given their technological choices,and then for the technological choices given their optimal productiondecision functions. We are therefore looking for a subgame perfectNash or Stackelberg equilibrium in technological adoption as a functionof the following six industry parameters:

• m , the average or expected size of the market;• V (or s ), the variability or volatility in the level of demand;• H, the differential investment cost for a flexible technology;• b , the slope of the demand function;• x, the average-cost-minimizing level of production;• s, the minimum level of average variable production cost for an inflex-

ible technology, which is equal, by assumption, to the constant mar-ginal cost of production c for a flexible technology.

3. The Best-Reply Functions at theTechnological Choice Stage

We first characterize the second-stage production equilibria given thetechnological choices made at the first stage and then characterize thebest responses at the technological-choice stage.

16. Suppose firm 1 is inflexible and firm 2 is flexible; then (recall that c 4 s) q2 (q1 )4 (1/2b ) (a 1 s 1 bq1 ), and firm 1 will prefer to produce at capacity if in such a casethe equilibrium price is above s, rather than not produce and realize zero profits. Produc-ing and selling at capacity x means a price of p 4

12(a ` s 1 bx ), which is larger than

s if a . bx ` s, in which case q2 (x ) . 0. If both firms are inflexible, then producingand selling at capacity means a price of p 4 a 1 2bx, which is larger than s if a . 2bx` s. If both firms are flexible, then the Cournot equilibrium price with both firms produc-ing will be p 4

13a `

23 s, which is larger than s if a . s; hence our assumption. Moreover,

the assumption on a simplifies the analysis insofar as it implies that for any realizationof demand (a ), there is a unique Cournot Nash equilibrium in stage 2 in which bothfirms produce. Our main results are not fundamentally changed if our assumption ona is relaxed, but then the analysis involves selecting among multiple equilibria and mustrely on numerical analysis.


3.1 The Expected Profits given the Outcomeof the Technological-Choice Stage

Since the flexibility variable g can take only two values, namely ƒ andi, we must base our analysis on the explicit consideration of the fourpossible technological configurations that may arise at the end of thefirst stage of the game. Under our assumption on a , we can characterizethe equilibrium at stage 2 by assuming that an inflexible firm producesand sells at capacity and that the competitor, if flexible, reacts optimallyto it. It is in that sense that commitment, under inflexibility, is said tobe ‘‘maximal.’’

We must first obtain the expected profits in stage 2 as a functionof the technological configuration outcome of stage 1. Suppose that atthe end of stage 1, the technological configuration is ( i, ƒ ), that is, onefirm inflexible and the other flexible. The flexible firm’s profit functionis given by

P f(a; i, ƒ ) 4 [a 1 s 1 b (q i ` q f )]q f 1 H, (1)

and the best response of the flexible firm is

q f(qi ) 41

2b(a 1 s 1 bqi ). (2)

The inflexible firm produces and sells the quantity q i 4 x. Hence thesecond-stage equilibrium is

(q i, qf ) 4 1x,1

2b(a 1 s 1 bx )2. (3)

Substituting these values in the profit function (1) and denoting by EP g

(g ′, g 9), g [ $g ′, g 9}, the expected profit of the firm with technology gwhen the technological choices are (g ′, g 9), we get the reduced-formprofit functions:

EP f(i, ƒ ) 41

4b[V ` (m 1 s 1 bx )2 ] 1 H, (4)

EP i( i, ƒ ) 412x (m 1 s 1 bx ). (5)

Suppose that at the end of stage 1, the technological configurationis ( i, i ), that is, both firms are inflexible. Then both firms produce andsell quantity q 4 x for all values of a , and we obtain

EP i( i, i ) 4 x (m 1 s 1 2bx ). (6)

Suppose finally that at the end of stage 1, the technological con-


figuration is (ƒ, ƒ), that is, both firms are flexible. Then the second-stage Cournot equilibrium will be the standard Cournot equilibriumwith linear demand and constant marginal cost of production, that is,q f 4 (1/3b )(a 1 s ). Therefore

EP ƒ(ƒ, ƒ) 41

9b(m 1 s )2 `

19b

V 1 H. (7)

It is informative to derive an equivalent form of the reduced ex-pected-profit function by defining a variable d as follows:

d [ x 11

3b(m 1 s ),

that is, the difference between the capacity x and the Cournot equilib-rium quantity (1/3b ) (m 1 s ) when both firms are flexible and a isequal to its expected value m . The value (1/3b )(m 1 s ) may be calledthe average equivalent Cournot quantity. Our assumption on a impliesthat (1/6b )(m 1 s ) $ d . Therefore d [ 1(1/3b )(m 1 s ) # d # (1/6b )(m 1 s ) [ d . We can therefore write the expected profits (4), (5), (6)as follows [with EII f (ƒ, ƒ) still given by (7)]:

EP ƒ( i, ƒ ) 41

9b(m 1 s )2 `

14b

V 113 d (m 1 s ) `

14 bd 2 1 H, (4′)

EP i( i, ƒ ) 41

9b(m 1 s )2 `

16 d (m 1 s ) 1

12 bd 2, (5′)

EP i( i, i ) 41

9b(m 1 s )2 1

13 d (m 1 s ) 1 2bd 2, (6′)

where (1/9b ) (m 1 s )2 is the profit of each firm (over variable costs)at the certainty-equivalent Cournot equilibrium.

We can now analyze the technological-choice stage and character-ize the best response function BR(⋅) of a firm for different choices oftechnological flexibility by its competitor.

3.2 The Best Response to Inflexibility

Suppose that a firm’s competitor has chosen an inflexible technology.Under our assumption on a , both EII f ( i, ƒ ) and EII i ( i, i ) are positive,and therefore the firm will never decide as a best response to stay outof the market. Flexibility is the firm’s best response iff EP f

(i, ƒ ) $ EP i (i, i ), that is to say, from (4) and (6),

ƒ 4 BR(i ) ⇔ 14b

[V ` (m 1 s 1 3bx )2 ] 1 H $ 0, (8)

or equivalently, from (4′) and (6′),


ƒ 4 BR(i ) ⇔ 14b

V `94 bd 2 1 H # 0. (8′)

The first term of the above inequality (8′) depends on the volatility ofthe market, whereas the second term is an increasing function of theabsolute value of d , the gap (either positive or negative) between thecapacity production level x and the average equivalent Cournot pro-duction level. Clearly, if the capacity production level x is equal to(1/3b )(m 1 s ) and the volatility V of the market size is equal to 0, thenas far as the profit over variable costs is concerned, the firm would beindifferent between a flexible technology and an inflexible technology:the best second-stage production level with a flexible technology isequal to (1/3b ) (m 1 s ), the production level at which an inflexiblefirm would be committed. As the volatility of the market increases, theflexible technology becomes relatively more profitable, since it allowsthe firm to adjust its second-stage production level to the observedlevel of demand. The term (1/4b )V is the gain coming from this moreadapted response.

From (8) and (8′), we may characterize the conditions under whichflexibility is a best response to inflexibility as follows:

Proposition 1: Flexibility is a best response to inflexibility, f 4 BR( i ),when the sum of the volatility effect (1/4b )V . 0 and the capacity effect94bd 2 . 0 is larger than the differential fixed cost H, a condition that is morelikely satisfied:

• the larger the variance V of the market size is,• the larger the gap between the capacity level x and the average equivalent

Cournot quantity (1/3b )(m 1 s ) is,• the smaller the differential cost H of the flexible technology is.17

3.3 The Best Response to Flexibility

Suppose now that a firm’s competitor has chosen a flexible technology.Under our assumption on a , both EP f (ƒ, ƒ) and EP i (ƒ, i ) are positive,and therefore the firm will never decide as a best response to stay out

17. From (8) and (8’), we could add:

• the larger the capacity factor (m 1 s 1 3bx )2 is,• the further (smaller or larger) the expected size m of the market is from 3bx ` s,• the further the average variable cost s is from m 1 3bx,• either the further the absolute value of the slope of the inverse demand function b is from (1/3x )

(m 1 s ) ` (1/9b 2 ) (2H), when H #34 (m 1 s )x, or the smaller the absolute value of the

slope of the inverse demand function is, otherwise.


of the market. Flexibility is the firm’s best response iff EP f (ƒ, ƒ) $EP i (ƒ, i ), that is to say, from (5) and (7),

ƒ 4 BR(ƒ) ⇔1

9b[V ` m (m 1 2s 1

92 bx ) `

12 (3bx ` 2s )(3bx ` s )] 1 H $ 0, (9)

or equivalently from (5′) and (7),

ƒ 4 BR(ƒ) ⇔ 19b

V `12bd 2 1

16d (m 1 s ) 1 H $ 0. (9′)

Again the difference in profits permitted by a flexible technology overan inflexible one can be split into two terms. The first term, (1/9b )V,comes from a better adaptability to changing market conditions. Thesecond one, 1

2bd 2 116d (m 1 s ), comes from the gap (either positive or

negative) between the capacity production level x and the productionlevel of the leader in a Stackelberg production game where both firmsare flexible and the market volatility is zero.18 Suppose that x 4 (1/2b )(m 1 s ); then it is clearly better to be inflexible than flexible, sinceadopting a flexible technology would drive the firms to the Cournotequilibrium in the production subgame, which is less profitable thanthe Stackelberg equilibrium for the leader. Hence, the capacity effectis working against flexibility when x is close to (1/2b ) (m 1 s ).19

We can conclude as follows:

Proposition 2: Flexibility is a best response to flexibility, ƒ 4 BR(ƒ),when the sum of the volatility effect (1/9b )V, permitted by a better second-stage adjustment with a flexible technology, and the capacity effect 1

2bd 2 116d (m 1 s )—which is positive if x , (1/3b ) (m 1 s ), or negative if (1/3b )(m 1 s ) , x , (1/2b ) (m 1 s )—is larger than the differential fixed costH, a condition that is more likely satisfied:

• the larger the variance V of the market size is,• the smaller the capacity production level x of an inflexible technology is,• the smaller the differential cost H of the flexible technology is.20

18. Let us denote this second term by Z(d ); it is positive [negative] if x , [.] (1/3b )(m 1 s ), and it is minimized at d 4 (1/6b ) (m 1 s ), that is, at x 4 (1/2b )(m 1 s ), theleader’s production level in the second stage when both firms are flexible and the marketvolatility is zero.

19. Which, by our assumption on a , is the upper bound of the admissible values.20. From (9) and (9’), we could add:

• the further (either smaller or larger) the expected size of the market m is from 94bx ` s,

• the further the average variable cost s is from m 194bx,

• the smaller the absolute value of the slope of the inverse demand function b is.


From the best responses, we can characterize the Nash and Stack-elberg equilibria in technological flexibility.

4. Technological-Flexibility Equilibria

From expressions (8), (8′), (9), and (9′), the characterization of Nashand Stackelberg equilibria in terms of the six parameters of the modelwill involve some intricate conditions, which we will illustrate graphi-cally in particular two-dimensional spaces. Nevertheless, it seems pref-erable to express the characterizing conditions in their general formsand let the readers represent them along their preferred coordinates.

4.1 Simultaneous-Move Equilibria

Suppose that the firms move simultaneously in the technology-choicestage. From the characterization of best-reply functions in Propositions1 and 2, we can directly derive the following proposition on Nashequilibria.21

Proposition 3: The subgame perfect Nash equilibrium of the gamewhere at the technology-choice stage both firms move simultaneously can becharacterized as follows:

(a) both firms choose inflexible technologies if

H $ max h 14b

V `94 bd 2,

19b

V `12 bd 2 1

16 d (m 1 s )j,

that is, if neither (8) [or equivalently (8′)] nor (9) [or equivalently (9′)] issatisfied—domain I in Figure 1 (where s 4 ÏV) and in Figure 2;

(b) both firms choose flexible technologies if

H # min h 14b

V `94 bd 2,

19b

V `12 bd 2 1

16 d (m 1 s )j,

that is, if both (8) [(8′)] and (9) [(9′)] are satisfied—domain II in Figure 1and in Figure 2;

21. We restrict our attention to pure strategies. The reader should be aware of thepossibility of mixed-strategy equilibria, but their analysis would not improve our under-standing of the phenomena under study.


FIGURE 1. THE NASH (g 1*, g 2*) AND STACKELBERG (gL *, gF *)EQUILIBRIA IN (m , s ) SPACE FOR b 4 x 4 1, s 4 0.2, H 4 0.05


FIGURE 2. THE NASH (g 1*, g 2* ) AND STACKELBERG (gL *, gF *)EQUILIBRIA IN (d , H ) SPACE FOR m 4 5, s 4 0.7, b 4 1, s 4 0.2


(c) one firm chooses the flexible technology and the other the inflexibletechnology if

14b

V `94 bd 2 $ H $

19b

V `12 bd 2 1

16 d (m 1 s ),

that is, if (8) [(8′)] is satisfied but not (9) [(9′)]—domain III in Figure 1 andin Figure 2;

(d) both firms choose the same technology, either flexible or inflexible,if

14b

V `94 bd 2 # H #

19b

V `12 bd 2 1

16 d (m 1 s ),

that is, if (8) [(8′)] is not satisfied but (9) [(9′)] is—domain IV in Figure 1and in Figure 2. M

Proposition 3 shows that flexible and inflexible firms may verywell coexist in a Nash equilibrium. This will be the case if the differen-tial investment cost H of flexible technologies is neither too small nortoo large, in which case one firm adopts a flexible technology and theother an inflexible one. We will see later that the most profitable tech-nology is not always the same. This clearly means that the coexistenceof FMS technologies and traditional (inflexible or less flexible) technolo-gies is not pathological: there are no compelling reasons why the equi-librium should be symmetric, and such an asymmetric configurationis quite compatible with choices made by rational and competent man-agers in a symmetric strategic environment. We will go one step furtherin understanding that situation by looking at the Stackelberg equilibriain long-run technological-flexibility choices.

4.2 Sequential-Move Equilibria

Suppose that the firms move sequentially in the technology-choicestage. The Stackelberg equilibrium is a reasonable equilibrium conceptfor long-run decisions. Moreover, although the endogenous emergenceof a leader-follower market structure is not modeled here, it should beunderstood that such a structure may very well have emerged from aprevious stage even if firms were in a completely symmetric and similarposition at the beginning of that previous stage.22 Let us characterize

22. See Daughety and Reinganum (1994) for one such model. See also Boyer andMoreaux (1987) for a model in which firms agree on a distribution of roles. Assumingthat firms choose their technologies in a sequential move context is but one way togenerate an asymmetry between firms. There are of course other ways to do it. Ourobjective in the present paper is first to show that asymmetric technology equilibria could


the optimal technological choice of the first-mover firm. To achievethat, we must compare the leader’s profit for each choice of g [ $ i, f}given that the second mover will react according to its best responsecharacterized in Propositions 1 and 2.

When inflexibility is a dominant strategy, that is, when neither(8) [(8′)] nor (9) [(9′)] is satisfied, then clearly both the first mover andthe second mover choose the inflexible technology. When flexibility isa dominant strategy, that is, when both (8) [(8′)] and (9) [(9′)] are satis-fied, then clearly both the first mover and the second mover choosethe flexible technology.

When inflexibility is the best response to flexibility [i 4 BR(ƒ)]whereas flexibility is the best response to inflexibility [ƒ 4 BR( i )], thefirst-mover must compare EP i ( i, f ), which he gets when choosing aninflexible technology, with EP f (ƒ, i ), which he gets when choosing aflexible technology. Hence, from (4) and (5), he will choose a flexibletechnology if

14b

[V ` (m 1 s 1 bx )2 ] 112 x (m 1 s 1 bx ) 1 H $ 0, (10)

that is, from (4′) and (5′), if

14b

V 112 d (m 1 s ) `

34 bd 2 1 H $ 0. (10′)

When inflexibility is the best response to inflexibility [i 4 BR(i )]whereas flexibility is the best response to flexibility [ƒ 4 BR(ƒ)], thefirst mover must compare EP i (i, ƒ ), which it gets when choosing aninflexible technology, with EP f (ƒ, i ), which it gets when choosing aflexible technology. Hence, from (6) and (7), it will choose a flexibletechnology if

19b

[V ` (m 1 s )2 ] 1 x(m 1 s 1 2bx ) 1 H $ 0, (11)

that is, using (8′), if

19b

V `13 d (m 1 s ) ` 2bd 2 1 H $ 0. (11′)

Lemma 1: If i 4 BR( i ) and f 4 BR(ƒ), then EP i ( i, i ) . EP f (ƒ, ƒ).

result from the strategic competition between symmetric firms. Introducing a Stackelbergcontext will allow us to obtain a unique equilibrium for each industry configuration andthen to perform comparative statics analyses.


Proof: We must show that if (9) is met but not (8) (domain IV in thefigures), then (11) is not met and the leader always choose an inflexibletechnology. We have i 4 BR( i ) and ƒ 4 BR(ƒ) only if

4bH 1 (m 1 s 1 3bx )2 . 9bH 1 m (m 1 2s 192 bx )

112 (3bx ` 2s )(3bx ` s ),

which holds iff

15bH `32 bx(m 1 s 1 3bx ) $ 0, (,.1)

which holds only if m 1 s 1 3bx $ 0. Now, (11) fails to be satisfiedwhenever i 4 BR( i ) and ƒ 4 BR(ƒ) if

4bH 1 (m 1 s 1 3bx )2 , 9bH ` 9bx(m 1 s 1 2bx ) 1 (m 1 s )2,

which holds iff

5bH ` 3bx(m 1 s 1 3bx ) $ 0. (,.2)

Therefore, $ i 4 BR( i ), ƒ 4 BR(ƒ)} ⇒ (,.1) ⇒ $m 1 s 1 3bx . 0} ⇒(, .2) ⇒ $(11) fails to be satisfied whenever i 4 BR( i ) and ƒ 4 BR(ƒ)}.

Q.E.D.

We may conclude as follows, denoting by EPL (g , g ′ ) and EPF (g ,g ′ ) the expected profits of the first mover (leader) and of the secondmover (follower) when the leader chooses a technology g and the fol-lower a technology g ′:

Proposition 4: The subgame perfect Stackelberg equilibrium (gL *,gF * ) can be characterized as follows:

(a) both firms choose inflexibility and their expected profits are similareither if

H $ max h 14b

V `94 bd 2,

19b

V `12 bd 2 1

16d (m 1 s )j,

that is, if neither (8) [(8′)] nor (9) [(9′)] holds—domain I in Figure 1 and inFigure 2—or if

14b

V `94 bd 2 # H #

19b

V `12 bd 2 1

16 d (m 1 s ),

that is, if (9) [(9′)] holds but not (8) [(8′)] (implying, by Lemma 1, that (11)[(11′)] is not met)—domain IV in Figure 1 and in Figure 2;


(b) both firms choose flexibility and their expected profits are similar if

H # min h 14b

V `94 bd 2,

19b

V `12 bd 2 1

16 d (m 1 s )j,

that is, if both (8) [(8′)] and (9) [(9′)] hold—domain II in Figure 1 and inFigure 2;

(c) the first mover will be flexible and the second mover inflexible if

max h 14b

V `94 bd 2,

14b

V 112d (m 1 s ) `

34 bd 2j

$ H $1

9bV `

12 bd 2 1

16 d (m 1 s ),

that is, if (8) [(8′)] and (10) [(10′)] hold but not (9) [(9′)]—domain III.A inFigure 1 and in Figure 2; first-movership is the preferred position;

(d) the first mover will be inflexible and the second mover flexible if

14b

V`94 bd 2$H

$max h 19b

V`12 bd 21

16 d (m1s ),

14b

V112 d (m1s )`

34 bd 2j,

that is, if (8) [(8′)] holds but neither (9) [(9′)] nor (10) [(10′)]—domain III.Bin Figure 1 and in Figure 2; first-movership is the preferred position.

We immediately obtain the following corollaries.

Corollary 1: The expected profits of the first mover are never lowerthan the expected profits of the second mover, whatever the equilibrium con-figuration of technologies. Hence, technological flexibility choices are strategicsubstitutes rather than complements.

Consider Figure 1 and Figure 2. The Nash equilibria and the Stack-elberg equilibria are the same in domains I and II; in III and IV, thereare two Nash equilibria: $( i, f ), (ƒ, i )} in III and $( i, i ), (ƒ, ƒ)} in IV; butonly one Stackelberg equilibrium, the one most favorable to the firstmover: ( i, f ) in III.B, (ƒ, i ) in III.A, and ( i, i ) in IV. Hence:

Corollary 2: A flexibility trap is present in industries in domain IV,that is, for which

14b

V `94 bd 2 # H #

19b

V `12 bd 2 1

16 d (m 1 s ).


Although (ƒ, ƒ) is a legitimate Nash equilibrium, both firms would make moreprofits if they were to choose inflexible technologies ( i, i ).

It is interesting and revealing to see how the predicted Stackelbergequilibrium evolves as the volatility and the expected size of the marketchange. The following two corollaries give two particular ‘‘paths oftechnological-flexibility adoptions.’’ The reader should be aware thatthe comparative statics analysis performed here is only suggestive of atechnological flexibility adoption process whose rigorous analysiswould require a dynamic framework. However, the comparative staticsanalysis is indicative of the kind of pressures that changes in the under-lying parameter generate.23

Corollary 3: Suppose that the expected size of the market is fixed atm in Figure 1; then, as the volatility s of demand increases, we obtain thefollowing sequence of predicted equilibria (gL *, gF *): (i, i), (i, ƒ ), (ƒ, i), (ƒ, ƒ).Hence for intermediate values of demand volatility, an increase in volatilityinduces the two firms to ‘‘trade’’ their technologies: the inflexible leaderswitches to a flexible technology, inducing the follower to switch from a flexibletechnology to an inflexible one.

Corollary 4: Suppose that the volatility of demand is fixed at s inFigure 1; then, as the expected size m of the market increases, we obtain thefollowing sequence of predicted equilibria (gL *, gF * ): (i, ƒ ), (i, i), (ƒ, i), (ƒ, ƒ).Again the two firms may wish to ‘‘trade’’ their technologies, but through astage in which both firms choose inflexible technologies.

Corollary 5: Suppose that d , the difference between the capacity leveland the average equivalent Cournot quantity, is fixed at d

ˆin Figure 2; then,

as the differential cost of the flexible technology H decreases, we obtain thefollowing sequence of predicted equilibria (gL *, gF * ): (i, i), (i, ƒ), (ƒ, i),(ƒ, ƒ). The flexible technology is adopted first by the second mover, and againthe two firms may wish to ‘‘trade’’ their technologies.

To represent different industries directly in Figure 1 as points in(m , s ) space, we must normalize the measurement of m and s . To doso, we can normalize the production levels q in terms of x and b byrewriting the demand function in a given industry as

23. An alternative interpretation would be that we consider only the long-run equilib-ria and not the transitional phases through which a long-run equilibrium is obtained.Another interpretation, suggested by a referee, would be that technologies depreciateinstantaneously with the change in parameters.


p 4p

bx 4abx 1 b

q L ` q F

bx 4 a 1 ( q L ` q F ).

In this way, b and x can be said to be normalized to 1. We obtainE( a ) 4 E(a )/bx and s ( a ) 4 s (a )/bx. Hence, changes in b or x in aparticular industry move the point representing it, in the (m , s ) space,on the ray from the origin through the original point: away from theorigin if bx decreases, and towards the origin if bx increases. Thosechanges have no effect on the boundaries of the different domains inFigure 1.

Let us consider three changes typically associated with the emer-gence of flexible technologies: a reduction in the minimum efficiencyscale x, a reduction in b , and a reduction in the differential cost offlexibility H. The first two changes represent increases in competitivepressures: the reduction in b implies an increase in demand elasticityat all production levels,24 while the reduction in x may be interpretedas softer entry conditions and therefore more aggressive competitionfrom the competitive fringe for our duopolists.

Corollary 6: Increasing competitive pressures, either through an in-crease in the price elasticity of demand or through a reduction of the mininumefficiency scale of production, is favorable to flexible technologies, but theadoption path depends on the relative position of the industry in (m , s ) space:

• for industries in I close to the border of III.A, one would predict a switchfrom (i, i) to (i, ƒ), that is, an increase in overall flexibility because of thesecond mover’s switch to a flexible technology;

• for industries in I close to the border of III.B, one would predict a switchfrom (i, i) to (ƒ, i), that is, an increase in overall flexibility due to the first-mover’s switch to a flexible technology;

• for industries in III.A close to the border of III.B one would predict a switchfrom (i, ƒ) to (ƒ, i), that is, a technology ‘‘trade’’ between the first moverand the second mover;

• for industries in III.B close to the border of IV, one would predict a switchfrom (ƒ, i) to (ƒ, ƒ), that is, an increase in overall flexibility due to thesecond-mover’s switch to a flexible technology;

• for industries located in I close to the border of II, one would predict a switchfrom (i, i) to (ƒ, ƒ), that is, an increase in overall flexibility because of aswitch by both firms to a flexible technology.

The following corollary on the impact of a reduction in the differ-

24. But of course the demand elasticity at every price level remains the same.


ential investment cost H of flexible technologies is expressed in termsof the distribution of industries in the (m , s ) space. A change in H hasno effect on the distribution of industries in (m , s ) space, but displacesthe boundaries of the different domains. If H decreases by dH, then allthe boundaries in Figure 1 move down vertically: (10) and (8) by 4 dH,and (9) by 9 dH. Suppose that the normalized minimum value of m isabove the value for which (8) intersects with the horizontal axis, say m$ 2.8. Then a reduction in the differential investment cost H of flexibletechnologies increases domain II, reduces domains I and III.A, but hasno effect on domain III.B. Hence:

Corollary 7: A reduction in the differential investment cost H of flexi-ble technologies unambiguously reduces the probability of observing equilib-rium configuration (i, i) and increases the probability of observing equilibriumconfiguration (ƒ, ƒ). Although the overall effect is favorable to the adoptionof flexible technologies, the specific effect on the probabilities of observingasymmetric equilibria (ƒ, i) and (i, ƒ) depends on the distribution of industriesin (m , s ) space. Assuming that the distribution is uniform over [(2.8, 0), (m ,s )] for the parameter values of Figure 1, then a reduction in H would notchange the probability of observing equilibrium configuration (i, ƒ) but wouldreduce the probability of observing equilibrium configuration (ƒ, i), henceincreasing the probability of (i, ƒ) relative to (ƒ, i).

5. Technological Flexibility and EntryDeterrence

Technological choices may be aimed at preventing entry. We will char-acterize in this section the circumstances under which the first movermay switch to a more (less) flexible technology in order to prevententry and those under which technology cannot be used to prevententry. Suppose that there exists a sunk cost of entry K independent ofthe investment cost of the technology that the entrant will eventuallychoose. We will continue to assume that H(i ) 4 0 and H(ƒ) 4 H . 0.

We must first characterize the level of profits a firm would obtainif it were able to prevent entry and act as a monopolist.

Proposition 5: The optimal technology choice of a monopolist can becharacterized as follows:

gM 4 hƒ if (1/4b )[V ` (m 1 s )2 ] 1 x(m 1 s 1 bx ) 1 H $ 0,i otherwise.

Proof: If g 4 i, then the monopolist inelastically puts on the marketthe quantity q 4 v 4 x and his profit is


EP M(i ) 4 E[(a 1 bx )x 1 sx ] 4 x(m 1 s 1 bx ). (12)

If g 4 ƒ, then the monopolist puts on the market the quantity q(a ) 4(1/2b )(a 1 s ) and realizes an expected profit of

EP M(ƒ) 4 E $[a 1 s 1 bq (a )]q(a )} 1 H 41

4b[V ` (m 1 s )2 ] 1 H.

(13)

The proposition follows from comparing the profit levels.Q.E.D. M

We can use (12), (13), the profit functions derived in Section 3,and Proposition 4 above to obtain the following propositions. In eachcase, the leader will consider switching from g , the flexible (inflexible)technology, to g ′, the inflexible (flexible) one, to enjoy monopoly profitswhen doing so can prevent entry, that is, when K and EP F (⋅, ⋅) satisfythe following two conditions, with g ≠ g ′:

EP F(g ′, BR(g ′ )) , EP F(g , BR(g )), (14)

K $ EP F(g ′, BR(g ′ )), (15)

that is, when the entrant’s profit decreases with the switch by the in-cumbent from g to g ′ and the entry cost K is at a proper level. Byassumption, K , EP F (g , BR(g )), that is, the original choice of theincumbent would not prevent entry.

Proposition 6: When ƒ 4 BR(ƒ) and ƒ 4 BR(i ) (domain II in Figure1), the leader chooses a flexible technology; he will switch to an inflexible oneif

V # 9bH 1 (m 1 s )2 ` 9bx (m 1 s 1 bx ), (16)

V # 1 (m 1 s )2 `185 bx(m 1 s ) 1

95 b 2x 2 (17)

provided that K is at an appropriate level.

Proof: The leader would consider switching to an inflexible technol-ogy if EP M ( i ) . EP L (ƒ, ƒ), which is condition (16), and if EP F (i, ƒ ), EP F (ƒ, ƒ), which is condition (17). Hence if those conditions aresatisfied, the leader will switch to an inflexible technology providedthat K satisfies (15) with g ′ 4 i. Q.E.D. M

Conditions (16) and (17) define the hatched subdomain of domainII in Figure 1; (17) is well above (16) and therefore does not appear onthe graph. This subdomain represents industries in which a leader (or


incumbent) will choose an entry-preventing inflexible technology, if theentry cost K is at an appropriate level, rather than the flexible technol-ogy he would choose otherwise.

Proposition 7: When i 4 BR(ƒ) and ƒ 4 BR(i ) (domain III in Figure3), the leader will never operate a switch in his technology, because doing socannot prevent entry.

Proof: If the leader has chosen a flexible technology, then it must bethe case that EP L (ƒ, i ) . EP L ( i, ƒ ); but recall that EP L (ƒ, i ) 4 EP F ( i,ƒ ) and that EP L ( i, ƒ ) 4 EP F (ƒ, i ); hence it is impossible for the leaderto prevent entry, since EP L (ƒ, i ) . EP L ( i, ƒ ) implies EP F ( i, ƒ) .EP F (ƒ, i ). And similarly for the case where the leader has chosen aninflexible technology. Q.E.D. M

Hence in those industries characterized by a (m , s ) in domainIII, a change in technological flexibility, either from ƒ to i or from i toƒ, cannot prevent entry.

Proposition 8: When i 4 BR(ƒ) and i 4 BR(i ) (domain I in Figure3), the leader chooses an inflexible technology; he would switch to a flexibletechnology if

V . 4bH 1 (m 1 s )2 ` 2bx(m 1 s 1 2bx ), (18)

m $ 3bx ` s (19)

provided that K is at an appropriate level.

Proof: Condition (18) is the condition under which EP M (ƒ) . EP L (i,i ), and condition (19) is the condition under which EP F (ƒ, i ) , EP F ( i,i ); therefore, if those conditions hold, the leader will switch to flexibilityprovided that K satisfies (15) with g ′ 4 ƒ. Q.E.D. M

Note that conditions (18) and (19) define the hatched subdomainof domain I in Figure 3; this subdomain represents industries in whicha leader (or incumbent) will choose an entry-preventing flexible technol-ogy, if the entry cost K is at an appropriate level, rather than the inflexi-ble technology he would choose otherwise.

6. Conclusion

Most studies of technological flexibility concentrate on the minimiza-tion of costs in a decision-theoretic context. We have shown in thispaper that those choices have important strategic implications that de-pend on market structure. More precisely, we have shown how techno-logical flexibility choices and equilibrium (both simultaneous and se-


FIGURE 3. THE NASH (g 1*, g 2* ) AND STACKELBERG (gL *, gF *)EQUILIBRIA IN (m , s ) SPACE FOR b4x41, s40.2, H43


quential) configurations in different industries depend on six industrialcharacteristics and on the strategic positioning of the firms, and howchanges in those characteristics are likely to affect the technological-flexibility configuration observed in a given industry and therefore thedistribution of those flexibility configurations in the economy as awhole. We abstracted from many activities and decisions behind a flexi-ble manufacturing system (such as product design and differentiation,organization, kanban integration, rapid mass data communications) byconsidering a single cost function with an explicit flexibility parameterand an explicit capacity parameter. However, we should stress that forany capacity level, more flexibility means in our model a reduction insetup costs, in minimum production runs, in inventories, and in vari-able production costs at all production levels, as well as speedier adjust-ments to changing market conditions (modeled through a random mar-ket demand), but requires a higher investment cost. In a strategiccontext, more flexibility may come at the expense of a commitment andpreemption strategy, whose credibility could rely on the inflexibilityto adapt to changing market conditions. We analyzed the balance be-tween the two strategies and determine the factors which tilt the bal-ance one way or another.

Flexible and inflexible technologies can coexist in an industrywhen the expected or average value m and the volatility s of the marketfall in a particular region of the parameter space, which we character-ized (domain III in Figures 1, 2, and 3). Low market volatility combinedwith intermediate market size favors inflexible technologies; large val-ues of either expected market size or market volatility favor flexibletechnologies; low or intermediate values of both market volatility andmarket expected size favor asymmetric choices of technological flexibil-ities.

Our results not only shed light on the underlying factors explain-ing the stylized fact described in the introduction (and its evolutionover time), namely that Japan invests significantly more in FMS tech-nologies than the United States, but also suggest explicit empirical hy-potheses to be tested with time-series data on an industry or with cross-sectional data on a set of industries. Those empirical hypotheses charac-terize the effect of variations in the relative investment costs of flexibleand inflexible technologies, in market size, in demand volatility, andin competitive market pressures on the equilibrium configuration (or‘‘adoption path’’) of technologies in different industries. Althoughmany of those effects might confirm one’s prior expectations, a signifi-cant subset are rather surprising: firms may wish to ‘‘trade’’ their tech-nologies; the increase in industry flexibility may sometimes come fromthe leader (dominant) firm and sometimes from the follower firm; in-


creases in market size may favor inflexible technologies; increases inexogenous competitive pressures, measured either as an increase indemand elasticity or as a decrease in the minimum efficiency scale ofproduction, will induce firms to switch, one at a time, from an inflexibletechnology to a flexible one, and although the leader (dominant) firmwill usually adopt the flexible technology first, there are nonpathologi-cal cases where it is the follower who adopts it first; in those lattercases, a further increase in competitive pressures will induce the firmsto trade their technologies, from ( i, ƒ ) to (ƒ, i ), before further increasesmake it profitable again for the follower to reverse again to a flexibletechnology; a reduction in the investment cost of flexible technologiesis favorable to the adoption of those technologies, but its specific effecton the probability of observing asymmetric equilibria (ƒ, i ) and ( i, ƒ )depends on the distribution of industries over the parameter spacedefined by market size and market volatility; firms in an industry mayfind themselves in an ‘‘excessive flexibility’’ trap; entry prevention maysometimes be achieved by flexible technologies and sometimes by in-flexible technologies, depending on the position of the industry in pa-rameter space. Hence, some prudence is required in predicting theemergence (and adoption) of technological-flexibility positions byfirms: strategic contexts differ significantly from decision-theoretic con-texts.25

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